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Mirrors > Home > ILE Home > Th. List > dfrnf | GIF version |
Description: Definition of range, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Ref | Expression |
---|---|
dfrnf.1 | ⊢ Ⅎ𝑥𝐴 |
dfrnf.2 | ⊢ Ⅎ𝑦𝐴 |
Ref | Expression |
---|---|
dfrnf | ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrn2 4637 | . 2 ⊢ ran 𝐴 = {𝑤 ∣ ∃𝑣 𝑣𝐴𝑤} | |
2 | nfcv 2229 | . . . . 5 ⊢ Ⅎ𝑥𝑣 | |
3 | dfrnf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
4 | nfcv 2229 | . . . . 5 ⊢ Ⅎ𝑥𝑤 | |
5 | 2, 3, 4 | nfbr 3895 | . . . 4 ⊢ Ⅎ𝑥 𝑣𝐴𝑤 |
6 | nfv 1467 | . . . 4 ⊢ Ⅎ𝑣 𝑥𝐴𝑤 | |
7 | breq1 3854 | . . . 4 ⊢ (𝑣 = 𝑥 → (𝑣𝐴𝑤 ↔ 𝑥𝐴𝑤)) | |
8 | 5, 6, 7 | cbvex 1687 | . . 3 ⊢ (∃𝑣 𝑣𝐴𝑤 ↔ ∃𝑥 𝑥𝐴𝑤) |
9 | 8 | abbii 2204 | . 2 ⊢ {𝑤 ∣ ∃𝑣 𝑣𝐴𝑤} = {𝑤 ∣ ∃𝑥 𝑥𝐴𝑤} |
10 | nfcv 2229 | . . . . 5 ⊢ Ⅎ𝑦𝑥 | |
11 | dfrnf.2 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
12 | nfcv 2229 | . . . . 5 ⊢ Ⅎ𝑦𝑤 | |
13 | 10, 11, 12 | nfbr 3895 | . . . 4 ⊢ Ⅎ𝑦 𝑥𝐴𝑤 |
14 | 13 | nfex 1574 | . . 3 ⊢ Ⅎ𝑦∃𝑥 𝑥𝐴𝑤 |
15 | nfv 1467 | . . 3 ⊢ Ⅎ𝑤∃𝑥 𝑥𝐴𝑦 | |
16 | breq2 3855 | . . . 4 ⊢ (𝑤 = 𝑦 → (𝑥𝐴𝑤 ↔ 𝑥𝐴𝑦)) | |
17 | 16 | exbidv 1754 | . . 3 ⊢ (𝑤 = 𝑦 → (∃𝑥 𝑥𝐴𝑤 ↔ ∃𝑥 𝑥𝐴𝑦)) |
18 | 14, 15, 17 | cbvab 2211 | . 2 ⊢ {𝑤 ∣ ∃𝑥 𝑥𝐴𝑤} = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} |
19 | 1, 9, 18 | 3eqtri 2113 | 1 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} |
Colors of variables: wff set class |
Syntax hints: = wceq 1290 ∃wex 1427 {cab 2075 Ⅎwnfc 2216 class class class wbr 3851 ran crn 4453 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-v 2622 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-br 3852 df-opab 3906 df-cnv 4460 df-dm 4462 df-rn 4463 |
This theorem is referenced by: rnopab 4695 |
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